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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem
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Median Age The table shows the median ages of the U.S. resident population for the years 2005 through 2013. (Source: U.S. Census Bureau)

Year 2005 2006 2007 2008 2009
Median age 36.2 36.3 36.5 36.7 36.8
Year 2010 2011 2012 2013
Median age 37.2 37.3 37.5 37.6

Spreadsheet at LarsonAppliedCaIculus.com

(a) Use Simpson’s Rule to estimate the average median age over the time period.

(b) A model for the data is

A   =   37.2     0.54 t   +   0.084 t 2   0.0031 t 3 for 5   t   13 , where A is the median age and t is the year, with t   =   5 corresponding to 2005. Use integration to find the average median age over the time period.

(c) Compare the results of parts (a) and (b).

(a)

To determine

To calculate: The average median age over the time period for the table that shows the median ages of the U.S. resident population for the years 2005 through 2013.

Year 2005 2006 2007 2008 2009 2010 2011 2012 2013
Median Age 36.2 36.3 36.5 36.7 36.8 37.2 37.3 37.5 37.6
Explanation

Given Information:

The table shows that the median ages of the U.S. resident population for the years 2005 through 2013.

Year 2005 2006 2007 2008 2009 2010 2011 2012 2013
Median Age 36.2 36.3 36.5 36.7 36.8 37.2 37.3 37.5 37.6

Formula used:

Simpson’s Rule:

If f is continuous on [a,b] and n is an even integer, then

abf(x)dx=(ba3n)[f(x0)+4f(x1)+2f(x2)+4f(x3)++4f(xn1)+f(xn)]

The average median can be find as,

Average Median age=1baabf(x)dx

Calculation:

Since, the given table shows the population from 2005 to 2013.

So, here a=5 and b=13

Now, use the table and substitute a=5,b=13 and n=8 in the Simpson’s formula as,

abf(x)dx=1353×8[36.2+4(36.3)+2(36.5)+4(36.7)+2(36.8)+4(37

(b)

To determine

To calculate: The average median age over the time period for the model of data A=37.20.54t+0.084t20.0031t2.

(c)

To determine

The comparison between the results obtained from the average median age by using table of data and the average median age over the time period using model of the data.

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